Question: $ D = \left[\begin{array}{r}4 \\ -2\end{array}\right]$ $ E = \left[\begin{array}{rr}-2 & -2\end{array}\right]$ Is $ D E$ defined?
Solution: In order for multiplication of two matrices to be defined, the two inner dimensions must be equal. If the two matrices have dimensions $( m \times  n)$ and $( p \times q)$ , then $ n$ (number of columns in the first matrix) must equal $ p$ (number of rows in the second matrix) for their product to be defined. How many columns does the first matrix, $ D$ , have? How many rows does the second matrix, $ E$ , have? Since $ D$ has the same number of columns (1) as $ E$ has rows (1), $ D E$ is defined.